research: shared-donor chains and overlapping target recovery, conditional kappa = 1.016422320428301e-3

[Dugongue]

Result

Adds research/e8-shared-donor-chains with

kappa = 1016422320428301 / 10^18 = 0.001016422320428301, approximately 0.3070561% above #401 at 7844f72c6051a4408284c3092490687061675c1a.

The snapshot is frozen through #401, including the downloaded #376 head 7aa9d9f0247bb60c0519704bbca18990011e2a37. This is a comparison against that snapshot, not against later submissions. The changed finite supplier, reserve bridge, exact moments and outer assembly are rebuilt together; independent gains are never added.

New mechanism: nested shared-donor gauges

#401 required complete role disjointness between response-gauge groups. That restriction is stronger than the algebra needs. If pivots are excluded from every donor set, the entrance matrix still has the form S = I + N with N^2 = 0, even when donor helpers appear in several groups. Consequently the literal shears commute and the inverse is explicit.

The physical restriction is handled constructively: each reused donor's assigned entrance spaces form a nested chain. The entrance gates are scheduled in increasing dimension, so the actual donor register visits that chain once, with every rank difference and later climb charged. No free joins, frame resets or hidden copies are assumed.

This expands the fixed witness from 119 to 194 pivot helpers, using 204 donors in 42 groups, with 36 donors shared by two groups and 37 distinct entrance spaces. All 484 literal entrance gates have unit numerators and denominators 1 or 2. Exact reconstruction checks all 1,021 raw columns, all gate-omission response leaks, and the actual nested paths.

Additional chronological target recovery

A new recovery uses donor target124 and destination125. If their prefix increments are u and u+v, respectively, the literal transformation is:

125 -= 124
124 receives u; 125 receives v
125 += 124

For arbitrary initial dirty values, the destination ends at a-b+v+(b+u)=a+u+v. This pair overlaps an inherited dependency from125 to172. Its setup must occur after the existing zero-frame setups; moving it across that dependency fails a negative control.

The old disjoint-role/sum heuristic is replaced by an exact chronological 120-column target-matrix identity, alongside the complete 1,021-column raw equality, both independent symbolic endpoint checks, common-frame checks, private erasure and phase admission. All 52 target shears have identity endpoint in forward and chronological inverse-adjoint orientations. The helper/source cleanup subword is unchanged.

Its paid delta is {1:+1,3:-1,4:-1,6:+1}, with unchanged rank mass and one extra scalar gate. The broader native prefix screen found this pair where a prior three-target basis screen required an unavailable compatible third target.

Incorporated public improvement

From #376 at7aa9d9f0247bb60c0519704bbca18990011e2a37, this also takes the single frame change for target201 += helper491/2: frame203 of rank8 becomes frame942 of rank7. Structural matching locates it at B1592 in the new word; raw equality and nesting are freshly checked. This contributes {1:-2,2:+1} and is credited public work, not our new mechanism.

The complete combined local delta versus #401 is {1:+150,2:-12,3:-46,4:+1,5:-7,6:-6}, with rank mass reduced by79. All 4,162 final frame transitions are nested.

Complete finite accounting

The phase-correct two-sided residual normalization from #401 is retained and freshly applied to all37 entrances. Both boundary monomials surround the complete old-response/main/cleanup invocation; they are not commuted through its scalar gates. Both orientations must restore every independent helper token and erase private work before that wrapper theorem applies.

Eight replicas per stage give residual inventory {4:120,5:216,7:72,8:1144,9:4696}. The explicit integer packing uses 738 width73 banks per stage, down from748, with 394 rank-one pads. All27,936 boundary adapters, inverse owner permutations, cover corrections and private workspace are charged.

Quantity	Value
Complete local invocations	72
Banks per stage	738
Live / physical stock	14,322 / 15,113
Paid child calls	321,066
Rank mass / deficit	1,043,010 / 2,496
Maximum child rank	48
Real scalar primitives across invocations	5,707,776
Required / retained row coefficient	15,445 / 20,161
Complex coarse lower bound	254364121477661453089 / 250000000000000000000000
Unchanged bit coarse lower bound	255384852701747 / 250000000000000000

The complex side binds. The new source-counted reserve bridge, two exact rational moment engines and six fixed ordinary bootstrap levels satisfy all 47 strict outer inequalities at the displayed rational kappa.

Reproduction and validation

python -B research/e8-shared-donor-chains/verify.py /absolute/new/output

Default requirements are Python3.11+ and GCC/Clang C++17; CXX can be set. The replay is offline and needs no discovery solver or Lean. It reconstructs each transformation, checks dirty columns and both symbolic orientations, verifies 810 new/reflected local Gaussian phase edges and 18,944 residual address columns, then independently decodes all6,248 inverse routes, the bank schedule, profile and full invoice. Eight adverse structural mutations must fail, including omitted adapters, wrong inverses, aliased owners and underpriced padding.

Unchanged bit and old local-phase evidence is explicitly reused through exact source/witness bindings. --replay-bit optionally repeats the bundled bit component using its documented dependencies. No unchanged public PR pipeline is run by default. VALIDATION.json binds executed bytes; MANIFEST.json covers the complete upload. Discovery scores and finite solver optimality are not accepted as mathematical certificates.

Sources and retained research

• research: response-subspace gauges and two-sided normalization, kappa = 1.013310887459710e-3 #401: exact119-gauge baseline, arbitrary entrance-space two-sided normalization, mixed-bank architecture, source-bound receipts and portable framework.
• Conditional κ=0.0011208649522582786: split centres with complete signed admission #376: the newly incorporated frame edit, plus earlier helper/lifetime ancestry; the new head and older input lineage are distinguished in NOTICE.md.
• Title: Signed target-prefix cancellation on R781: conditional kappa = 0.001002494659629864 #398, E8 prefix restoration and heterogeneous helper banks: conditional kappa = 1.000220222889528e-3 #390: signed and positive target-prefix recovery.
• Conditional kappa 0.001001098432210898: signed-prefix donor gauges #396: entrance response kernels and the odd rank-one route generalized in research: response-subspace gauges and two-sided normalization, kappa = 1.013310887459710e-3 #401.
• Conditional κ = 9.5464544449649·10⁻⁴: Sussman's E8 complex unit re-timed by far piece relocation, in #375's unsplit B₃ (+0.042 % over #375) #378, Reduce E8 recursive cost with four coupled frame changes #365, Conditional κ = 9.54395132518163e-4: coupled complex retiming after #375 (+0.01588%) #374: retiming/frame ancestry; κ = 9.98243154231047e-4: nine-stage complex E8 (≈2^-9.9683; complex +14.036%, bit side pending) #377: nine-stage complex geometry, Clifford representatives, owner routing and finite invoice.
• κ = 9.95656966441037e-4: #370's E8 bit word in the nine-stage bridged word with #377's nine-stage complex E8 (above 2^-10; +2.94 % over #380) #381 and earlier Conditional κ=0.0011208649522582786: split centres with complete signed admission #376: unchanged bit composition.
• Conditional kappa 0.000998634658090835: finite bootstrap and exact coarse precision #387, Conditional kappa 0.001000220224856110: six fixed levels on #390 #392, Finite leaf bootstrap above PR182: conditional kappa = 0.000660627334074291 #185: ordinary bootstrap/slack; Conditional κ = 9.98634656132003·10⁻⁴: #377's nine-stage E8 complex component with our #378 re-timed E8 unit, paired with our #382 bit word (+0.039 % over #382) #384, Conditional κ = 7.64230861320245e-4: the paired-cube bit word at cube size p = 10 (h = 20, m = 100), with a new centre-sharing complex supplier (+1.21% over #314) #315, Paired cubes and shared completed cores for conditional kappa = 4.609169e-4 #144, Semantic precision and bulk resampling for a conditional 1099/10^8 > 2^-17 bound #23: exact engine lineage.
• Conditional kappa 0.001002683712533373: R781 heterogeneous banks and padding fusion #400, Conditional κ = 1.000385913854264·10⁻³: bank padding fused into three new free sink helpers on #390's E8 complex program (+0.0166 % over #390) #394, Conditional kappa 0.001001256225581264: signed prefixes and padding fusion #395, Conditional κ = 1.002661106042177·10⁻³: our #394 padding fusion on #398's R781 signed-prefix E8 complex program (+0.0166 % over #398) #399: reviewed heterogeneous-bank/prewhitening/padding-fusion routes. Their distinct cleanup bracket is not added; the attempted compatibility construction and its unpriced tradeoff are recorded.

Further source ancestry and original author/license notices remain intact. Research notes include native search counts, the sufficient shared-donor and chronological-recovery arguments, structural DAG matches, failed bounded screens and unpriced alternatives. The fresh DAG retrieval is bounded and mostly source-only, not a new Lean certification or an exhaustive theorem-library audit.

Scope

This remains a finite conditional construction, not an unconditional integer-multiplication theorem or Lean certificate. Inherited all-size compressed-window lift, arbitrary-correlated dirty-bank semantics, recursive compiler, ordinary-call/fixed-tape/restored-row, precision and analytic interfaces remain assumptions. We do not claim to invent linear transvections, establish literature-wide novelty or prove global optimality. General acyclic pivot/donor dependencies and an entrance-preserving prewhitening/fusion combination are possible next routes; neither is assumed in this result.

[Mister-Meeseeks]

Thank you @Dugongue. This contribution is now retained on main in the scoped 100-PR review and contributor credits.

Contribution: Shared-donor chains and overlapping target recovery.

Review and limits: Uses a generalized chronological/native contract. Legacy gx E4/E5 acceptance is not inferred; exact dirty-word and phase/bank checks are assessed within this source's stated interface.

The complete original source, notices and relevant execution evidence are retained through the pinned source archive. The review distinguishes fresh local runs, source-bound historical Linux CI and prior exact-head review; it does not claim 100 fresh full mathematical audits. Closing this historical PR as reconciled. This preserves parallel and cumulative credit without replacing the selected frontier or promoting priced research targets to completed constructions.

Reviewed head: 12f488264cdd74a3b60d0fb534f4955792384f5c. Publication: 4040a0f46d67b41c36a472a7cd4af791fd4c36cc.